To Download BBT as an E-Book.

To obtain a copy of our graduate level real analysis textbook,
REAL ANALYSIS, Bruckner, Bruckner & Thomson




NO LONGER USING WebDeliverySolutions.com ...Just go to our web site for download instructions.

To Download TBB as an E-Book.

Trying out a new way of getting our PDF files to the end user.

So this is just an experiment. To purchase a E-book version (really just a huge PDF
file) of our undergraduate real analysis textbook:

ELEMENTARY REAL ANALYSIS by Bruckner, Thomson, & Bruckner.

This is a large PDF file intended for viewing (in landscape) on a laptop or computer screen. All references are hyperlinked. Let us know if it doesn't work for your Kindle or IPad2 and we will design something appropriate.

The paging is different from the paperback version, but all references and numbering is otherwise the same.

NO LONGER USING WebDeliverySolutions.com. JUST GO TO OUR WEB SITE.

NEW BOOK--MATHEMATICAL DISCOVERY

We are preparing a new book on Mathematical Discovery. It is intended for students with just the usual high school background in mathematics (algebra and geometry) who wish to learn how mathematicians discover new methods to attack problems. It is based on a course Andy Bruckner gave to students at UC Santa Barbara for non math majors, covering interesting topics that require less background preparation than the usual university-level courses would need.

There will be (as usual) FREE PDF FILES as well as a trade paperback version. The first chapter on Tilings discusses the famous problem of "squaring the rectangle," i.e., tiling a rectangle with a number of unequal sized squares. You can download this one now.

Riemann integral indeed?

For fans of this BLOG and the DRIP crusade (yea, both of you) it may appear that I have gone to the dark side since my two most recent papers are studies of properties of the Riemann integral:

• B S Thomson, "Monotone convergence theorem for the Riemann integral", American Math Monthly, Vol. 117, no. 6, June-July 2010.
  

• B S Thomson, "Characterizing the indefinite Riemann integral", Real Analysis Exchange, 2010 [to appear].
  

You can download preprints from here and here respectively. I hope you will (both) grant me absolution if I promise to return to a polemic on this unfortunate integral. Please write.

FREE CALCULUS BOOK

The Calculus Integral [Beta0.2],
B S Thomson,
ClassicalRealAnalysis.com (2009).

Download a free PDF file from http://www.youpublish.com/files/23072.


Well "calculus book" doesn't quite describe it. It is an account of integration theory on the real line that starts in the initial chapters with the "calculus integral" (i.e., the original integral of Newton) and carries the development as far as the integrals of Lebesgue and Henstock-Kurzweil.

If you are, however, teaching a calculus course and have ever considered dropping the Riemann integral and its ugly step-sister the improper Riemann integral from the syllabus, then you might want to use the first three chapters as a basis for the teaching of the basic integral of the calculus.

A trade paperback version will be available shortly.

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #5

It clears up the mystery of the popcorn function.


Suppose a function is zero at every irrational point. What is its integral? Well, certainly, the points left out are insignificant. Not merely a set of measure zero but even countable. Such a set plays no role in determining the integral so your function can have any values on the rationals, or even remain undefined on the rationals. The simple answer is that your function is integrable on every interval with a zero integral.

Oh, what? Sorry? You have only learned the Riemann integral. Alas, the answer now is entirely different. Now the function must be defined at these missing rational points and the answer depends on how you define them. If the resulting function is integrable then certainly the value of the integral is zero, but it may or may not be integrable. Let xn be a listing of all the rationals and let your function be defined to be f(xn)=cn and with f(x)=0 at x irrational. What is a necessary and sufficient condition for f to be integrable [i.e., integrable in the dumb Riemann sense]? That's a tough question, but one that is not particularly important.


In 1875, K. J. Thomae discovered the now-famous example of a function of this kind that is continuous at all the irrationals and discontinuous at the rationals. This function has many names: the modified Dirichlet function, Thomae function, Riemann function, raindrop function, ruler function, and popcorn function.



His example is a nice curiosity in the study of continuous functions. But it is usually presented to students of integration theory as example of a seriously discontinuous function that is integrable. The student gets the impression that it is important to have continuity, that discontinuities must be controlled, that without proper configuration of the values of a function the integral is badly affected, and that integration theory has its mysteries. That's good teaching?

If we drop the Riemann integral then the popcorn function would not be mentioned in the context of integration theory and can return to its proper place in the study of continuity.

Is there a function continuous at every rational and discontinuous at every irrational?
Is there a function discontinuous at every rational and continuous at every irrational?

The answer to the first question is "no" and the answer to the second question is "popcorn."

Free books aren't free?

One of our correspondents expressed the view that our free books didn't seem,
to him at any rate, to be free. Presumably because YouPublish asks for $1.00 to download PDF files.

But all of our books are fully viewable and searchable on Google Book Search, which you can try out below.

In addition most of the chapters of TBB (for example) can be downloaded without charge from our web site page.
The other issue, I suppose, is whether the $1.00 charge on YouPublish is irritating. Irritating, I mean, in the sense that your free Lone Ranger Decoder Ring (only $1.00 shipping and handling) was irritating when you were 12 years old.

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